interval category

It is well-known that there are infinite possible intervals, even if one confines one's view to a single octave. This includes the rational intervals of Just Intonation and the irrational intervals of EDOs, EDONOIs, and other systems. However, it is helpful to consider interval categories, which there may be a finite number to consider.

The Schulter System

Margo Schulter describes her system for categorizing intervals in Regions of the Interval Spectrum, which begins:

"In naming categories of intervals, or regions of the spectrum in which they are found, there may be many valid and desirable schemes reflecting the diversity of viewpoints and styles to be found in world musics. What I describe here is merely one possible solution, and one influenced by my own musicmaking experience and philosophy which seeks an equitable and inclusive balance between intervals at or near simple integer ratios, and those having a more complex or active nature."

Schulter proposes the following categories and gives a tentative range of cents values for intervals that fall within those categories. In Regions, she points out, "A main caution is that the borders are inevitably 'fuzzy,' so that one region shades into another and suggested values in cents are more illustrative than definitive."

Interval Category	Associated Ranges of Cents Values (borders are "fuzzy")
Pure Unison (1:1)	0
Commas	0-30
Dieses	30-60
Minor Seconds	60-125
*small	60-80
*middle	80-100
*large	100-125
Neutral Seconds	125-170
*small	125-135
*middle	135-160
*large	160-170
Equable Heptatonic	160-182
Major Seconds	180-240
*small	180-200
*middle	200-220
*large	220-240
Interseptimal (Maj2-min3)	240-260
Minor Thirds	260-330
*small	260-280
*middle	280-300
*large	300-330
Neutral Thirds	330-372
*small	330-342
*middle	342-360
*large	360-372
Major Thirds	372-440
*small	372-400
*middle	400-423
*large	423-440
Interseptimal (Maj3-4)	440-468
Perfect Fourths	468-528
*small	468-491
*middle	491-505
*large	505-528
Superfourths	528-560
Tritonic Region	560-640
*small	560-577
*middle	577-623
*large	623-640
Subfifths	640-672
Perfect Fifths	672-732
*small	672-695
*middle	695-709
*large	709-732
Interseptimal (5-min6)	732-760
Minor Sixths	760-828
*small	760-777
*middle	777-800
*large	800-828
Neutral Sixths	828-870
*small	828-840
*middle	840-858
*large	858-870
Major Sixths	870-940
*small	870-900
*middle	900-920
*large	920-940
Interseptimal (Maj6-min7)	940-960
Minor Sevenths	960-1025
*small	960-987
*middle	987-1000
*large	1000-1025
Equable Heptatonic	1018-1040
Neutral Sevenths	1030-1075
*small	1030-1043
*middle	1043-1065
*large	1065-1075
Major Sevenths	1075-1140
*small	1075-1100
*middle	1100-1120
*large	1120-1140
Octave less diesis	1140-1170
Octave less comma	1170-1200
Pure Octave (2:1)	1200

See: interval size measure, Gallery of Just Intervals

Pure unison (1:1)           0 cents
 
Commas                   0-30 cents        (Section 11)
 
Dieses                  30-60 cents        (Section 11)
 
Minor seconds          60-125 cents        (Section 5)
        small              60-80 cents
        middle             80-100 cents
        large             100-125 cents
 
Neutral seconds       125-170 cents        (Section 6)
        small              125-135 cents
        middle             135-160 cents
        large              160-170 cents
 
Equable heptatonic
(heartland range)     160-182 cents        (Section 12)
 
Major seconds
(or tones)            180-240 cents        (Section 4)
        small              180-200 cents
        middle             200-220 cents
        large              220-240 cents
 
Interseptimal
(Maj2-min3)           240-260 cents        (Section 9)
 
Minor thirds          260-330 cents        (Section 2)
        small              260-280 cents
        middle             280-300 cents
        large              300-330 cents
 
Neutral thirds        330-372 cents        (Section 3)
        small              330-342 cents
        middle             342-360 cents
        large              360-372 cents
 
Major thirds          372-440 cents        (Section 2)
        small              372-400 cents
        middle             400-423 cents
        large              423-440 cents
 
Interseptimal         440-468 cents        (Section 9)
(Maj3-4)
 
Perfect fourths       468-528 cents        (Section 7)
        small              468-491 cents
        middle             491-505 cents
        large              505-523 cents
 
Superfourths          528-560 cents        (Section 10)
 
Tritonic region       560-640 cents        (Section 8)
        small              560-577 cents
        middle             577-623 cents
        large              623-640 cents
 
Subfifths             640-672 cents        (Section 10)
 
Perfect fifths        672-732 cents        (Section 7)
        small              672-695 cents
        middle             695-709 cents
        large              709-732 cents
 
Interseptimal         732-760 cents        (Section 9)
(5-min6)
 
Minor sixths          760-828 cents        (Section 2)
        small              760-777 cents
        middle             777-800 cents
        large              800-828 cents
 
Neutral sixths        828-870 cents        (Section 3)
        small              828-840 cents
        middle             840-858 cents
        large              858-870 cents
 
Major sixths          870-940 cents        (Section 2)
        small              870-900 cents
        middle             900-920 cents
        large              920-940 cents
 
Interseptimal         940-960 cents        (Section 9)
(Maj6-min7)
 
Minor sevenths       960-1025 cents        (Section 4)
        small              960-987 cents
        middle            987-1000 cents
        large            1000-1025 cents
 
Equable heptatonic  1018-1040 cents        (Section 12)
(heartland range)
 
Neutral sevenths    1030-1075 cents        (Section 6)
        small            1030-1043 cents
        middle           1043-1065 cents
        large            1065-1075 cents
 
Major sevenths      1075-1140 cents        (Section 5)
        small            1075-1100 cents
        middle           1100-1120 cents
        large            1120-1140 cents
 
Octave less diesis  1140-1170 cents        (Section 11)
 
Octave less comma   1170-1200 cents        (Section 11)
 
Pure octave (2:1)        1200 cents